A. q Theory of Optimal Investment with Unobserved Shocks

We build on the original setting of Lucas and Prescott (Reference Lucas and Prescott1971) and Mussa (Reference Mussa1977) but, as an important point of departure, we allow for the unobserved persistent shock to enter the investment cost function. Here, we modify the dynamic investment model in Erickson and Whited (Reference Erickson and Whited2000), in which capital is the endogenous quasi-fixed factor and the unobserved persistent shock is another fixed factor that evolves exogenously following a dynamic process (e.g., a first-order Markov process). The value of the firm $ i $ at time $ t $ , from which the firm derives its optimal decision on investment to maximize the expected present value of the discounted flow of future profits, is given by

(1) $$ {V}_{it}=E\left[\sum \limits_{j=0}^{\infty}\left(\prod \limits_{s=1}^j{b}_{i,t+s}\right)\left[{\Pi}_{t+j}\left({K}_{i,t+j},{\varsigma}_{i,t+j}\right)-\psi \right({I}_{i,t+j},{K}_{i,t+j},{W}_{i,t+j},{\nu}_{i,t+j}\Big)|{\Omega}_{it}\right]. $$

Here, $ E\left[\cdot |{\Omega}_{it}\right] $ is the conditional expectation operator and $ {\Omega}_{it} $ denotes the information set available to firm i at time t; $ {K}_{it} $ is the capital stock available at the beginning of period $ t $ ; $ {I}_{it} $ is the investment and $ {b}_{it} $ is the firm’s discount factor at time $ t $ ; $ {\Pi}_t\left({K}_{it},{\varsigma}_{it}\right) $ is the per period profit function, increasing in $ {K}_{it} $ , with $ {\varsigma}_{it} $ being the shock to the profit function; $ \psi \left({I}_{it},{K}_{it},{W}_{it},{\nu}_{it}\right) $ is the investment cost function including both the cost of adjusting the stock of capital and the direct purchase or sale cost of investment, where $ {\nu}_{it} $ is an exogenous shock to adjustment cost. $ {W}_{it} $ denotes the vector of state variables other than capitals, which may include technology shock, demand and cost shocks, and other aggregate shocks.

The cost function $ \psi \left({I}_{it},{K}_{it},{W}_{it},{\nu}_{it}\right) $ is increasing in $ {I}_{it} $ , decreasing in $ {K}_{it} $ , and convex in the first two arguments. The shocks $ \left({\varsigma}_{it},{\nu}_{it}\right) $ and state variables $ {W}_{i,t} $ are observed by the firm at time $ t $ , but these shocks and some components of $ {W}_{it} $ may not be fully observed by the econometrician. Finally, note that any other variable factors of production in the profit function (e.g., labor or materials) have already been optimized following static optimization problems by the firm. Also, for ease of notation, other observed factors are implicit and suppressed. For estimation, we will decompose $ {W}_{it} $ into observed factors and the unobserved persistent shock. We will add these additional factors in our empirical investment equation specifications later.

The firm maximizes the expected present value of the future profits $ {V}_{it} $ , subject to the capital stock accumulation identity

(2) $$ {K}_{i,t+1}=\left(1-{d}_i\right){K}_{it}+{I}_{it}, $$

where $ {d}_i $ is the firm i’s capital depreciation. We then obtain the “marginal” $ {q}_{it} $ from $ \frac{\partial {V}_{it}}{\partial {K}_{it}} $ , which measures the benefit of adding an incremental unit of capital to the firm. The first-order condition for maximizing the value of the firm in equation (1), subject to equation (2), then yields

(3) $$ \frac{\partial \psi \left(\cdot \right)}{\partial {I}_{it}}\equiv {\psi}_I\left({I}_{it},{K}_{it},{W}_{it},{\nu}_{it}\right)={q}_{it}. $$

In the original setting of Lucas and Prescott (Reference Lucas and Prescott1971) and Mussa (Reference Mussa1977) (see also Erickson and Whited (Reference Erickson and Whited2000)), which is common in the literature, a firm’s unobserved persistent shock does not enter the investment cost function. Our main innovation is to incorporate an additional source of unobserved firm heterogeneity into the firm’s investment decision problem. We develop an empirical investment equation that aligns with the theoretical model and propose a consistent estimation procedure that accounts for this unobserved factor. Importantly, our framework allows the unobserved persistent shock to influence the optimal investment decision not only through the production function but also by affecting the cost of investment. We argue that this feature remains consistent with the neoclassical theory of investment. The first-order condition above (equation (3)) highlights that incorporating the unobserved shock into the investment cost function $ \psi \left(\cdot \right) $ is essential for the dependence of the optimal investment on the unobserved shock, given the marginal $ {q}_{it} $ . This is because the direct impact of this shock on the firm’s profit function $ {\Pi}_t(\cdot ) $ through its production function is already subsumed in $ {q}_{it} $ , and the unobserved shock only shows up in the optimal investment through $ \psi \left(\cdot \right) $ as in equation (3).

B. Empirical Model of Investment Equation

To develop an empirical framework, we now present an investment equation consistent with the firm’s optimal investment decision problem in equation (1). Write $ {W}_{it}\equiv \left({Z}_{it},{\omega}_{it}\right) $ where $ {Z}_{it} $ represents the observed state variables, which may proxy for firm heterogeneity and demand factors, and $ {\omega}_{it} $ denotes the unobserved persistent shock. We consider a class of investment cost functions, including the cost of adjusting the stock of capital, as

(4) $$ \psi \left({I}_{it},{K}_{it},{W}_{it},{\nu}_{it}\right)={K}_{it}\left[{\tilde{f}}_0\left({Z}_{it},{\nu}_{it},{\omega}_{it}\right)+{\tilde{f}}_1\Big({Z}_{it},{\nu}_{it},{\omega}_{it}\Big)\frac{I_{it}}{K_{it}}+\frac{\gamma_{it}}{2}{\left(\frac{I_{it}}{K_{it}}\right)}^2\right], $$

where, in particular, $ {\tilde{f}}_1\left({Z}_{it},{\nu}_{it},{\omega}_{it}\right) $ denotes the linear adjustment cost.

From equation (4), it is clear that the feature of the model that renders the investment function to depend on $ {\omega}_{it} $ is specifically due to the linear adjustment cost $ {\tilde{f}}_1\left({Z}_{it},{\nu}_{it},{\omega}_{it}\right) $ , which is a function of the shock $ {\omega}_{it} $ , not merely due to the investment cost function $ \psi \left({I}_{it},{K}_{it},{W}_{it},{\nu}_{it}\right) $ depending on $ {\omega}_{it} $ . For example, if we have $ {\tilde{f}}_1\left({Z}_{it},{\nu}_{it},{\omega}_{it}\right)={\tilde{f}}_1\left({Z}_{it},{\nu}_{it}\right) $ , the cost function still depends on $ {\omega}_{it} $ because of $ {\tilde{f}}_0\left({Z}_{it},{\nu}_{it},{\omega}_{it}\right) $ , but this additive adjustment cost does not enter the investment equation as we can see in equation (5). In the literature, it is also typically assumed that the adjustment cost parameter $ {\gamma}_{it} $ is constant across firms as $ \gamma $ (but it may vary by the time $ t $ ). Combining these, we obtain

(5) $$ \frac{\partial \psi \left(\cdot \right)}{\partial {I}_{it}}\equiv {\psi}_I\left({I}_{it},{K}_{it},{W}_{it},{\nu}_{it}\right)={\tilde{f}}_1\left({Z}_{it},{\nu}_{it},{\omega}_{it}\right)+\gamma \frac{I_{it}}{K_{it}}={q}_{it}. $$

This equation clearly indicates that $ {q}_{it} $ is dependent on the unobserved persistent shock $ {\omega}_{it} $ , unless the linear adjustment cost $ {\tilde{f}}_1\left({Z}_{it},{\nu}_{it},{\omega}_{it}\right) $ is independent of $ {\omega}_{it} $ . Finally, the above equation can be rewritten, as in the literature, yielding the investment equation for which now both $ {q}_{it} $ and $ {\omega}_{it} $ enter as factors of investment:

$$ {y}_{it}=\beta {q}_{it}-{f}_1\left({Z}_{it},{\nu}_{it},{\omega}_{it}\right), $$

where $ {y}_{it}=\frac{I_{it}}{K_{it}} $ , $ \beta =1/\gamma $ , and $ {f}_1\left({Z}_{it},{\nu}_{it},{\omega}_{it}\right)={\tilde{f}}_1\left({Z}_{it},{\nu}_{it},{\omega}_{it}\right)/\gamma $ . To develop a simple regression equation in line with the literature, we can let, for example,

$$ {\tilde{f}}_1\left({Z}_{it},{\nu}_{it},{\omega}_{it}\right)=-{Z}_{it}\tilde{\theta}-\tilde{\alpha}{\omega}_{it}+{\nu}_{it}. $$

We then obtain the familiar regression equation as anFootnote ³ extension of Erickson and Whited (Reference Erickson and Whited2000) (equation (6) below becomes their equation (6) if we set $ \theta =\alpha =0 $ ):

(6) $$ {y}_{it}=\beta {q}_{it}+{Z}_{it}\theta +{\alpha \omega}_{it}+{u}_{it}, $$

where $ \theta =\tilde{\theta}/\gamma $ , $ \alpha =\tilde{\alpha}/\gamma $ , and $ {u}_{it}=-{\nu}_{it}/\gamma $ . An important implication of this investment equation is that, if omitted in the regression, the unobserved persistent shock $ {\omega}_{it} $ is a potential source of endogeneity. It can be correlated with $ {q}_{it} $ , while $ {u}_{it} $ is the usual exogenous shock. Note that, for ease of notation, other observed factors in both the profit and cost functions are included in $ {Z}_{it} $ . These variables can be added to the empirical investment equation and may not create additional endogeneity problems once the omitted shock $ {\omega}_{it} $ is controlled for.

C. Interpretation of the Persistent Shocks

In this section, we set out our interpretation of the persistent shock $ \omega $ in the context of the firm investment, adjustment costs, and Tobin’s $ q $ literature. Investment adjustment costs are central to dynamic models of capital accumulation, as they determine the speed and efficiency with which firms respond to changes in economic conditions. Hayashi (Reference Hayashi1982) formalized the link between $ q $ and investment under convex adjustment costs, showing that marginal $ q $ governs optimal investment decisions in the presence of installation frictions. These costs arise because capital goods cannot be instantaneously installed without incurring inefficiencies, such as production disruptions or resource misallocation.

In dynamic investment models, unobserved persistent shocks, such as technological advancements or improvements in information efficiency, play a critical role in shaping firms’ investment behavior by influencing adjustment costs (Greenwood et al. (Reference Greenwood, Hercowitz and Krusell1997), Stiroh (Reference Stiroh2002), Fisher (Reference Fisher2006), and Kogan and Papanikolaou (Reference Kogan and Papanikolaou2014)). These shocks affect the marginal cost of capital adjustment, thereby altering optimal investment trajectories and the speed of capital accumulation. Enhanced information efficiency, for instance, reduces informational frictions and uncertainty, enabling firms to make more accurate and timely investment decisions. This improvement mitigates costs associated with misallocation, delays, and errors, ultimately fostering a more efficient allocation of resources and, in turn, firm productivity.

Similarly, positive technology shocks can lower the costs and time required to upgrade capital equipment or adopt new production technologies. For example, the diffusion of cloud computing and automation technologies has enabled firms to scale operations rapidly without incurring the high fixed costs traditionally associated with IT infrastructure upgrades. In manufacturing, the integration of advanced robotics has streamlined production processes, reducing downtime and adjustment costs during technology transitions. In the renewable energy sector, technological improvements in battery storage and solar panel efficiency have accelerated investment cycles, making capital upgrades less costly and more frequent.

Our interpretation of $ q $ aligns with the existing literature on measurement error in $ q $ . As shown in equation (5), $ q $ is on the right-hand side of the first-order condition and, in theory, it perfectly measures the marginal benefit of adding an incremental unit of capital to the firm. However, in practice, since it is unobserved and replaced with the average $ q $ , it is subject to measurement error and may not fully reflect, for example, a firm’s productivity variation stemming from intangibles, information asymmetries, or market inefficiencies.

Regarding how to handle this measurement issue in $ q $ , we depart from the existing literature by introducing unobserved persistent shocks $ \omega $ that affect adjustment costs on the left-hand side and can also capture, for example, firm productivity variation if it is not fully reflected in $ q $ . In our model, the unobserved persistent shocks effectively streamline the investment process and improve overall productivity. We incorporate them in the structural models for the firm’s optimization problem, but we test the theory based on the reduced-form model in equation (5). As shown in the next section, the empirical findings confirm that the unobserved persistent shocks are statistically significant and economically important in corporate investment decisions. The results stay robust even for the investment model with the total $ q $ that includes both physical and intangible capital. Failure to account for these unobserved persistent shocks may lead to biased estimates of investment dynamics and misinformed policy prescriptions. Incorporating such factors into investment models is therefore essential for accurately capturing the interplay between technological progress, firm behavior, and economic outcomes.

A. Modeling the Unobserved Persistent Shock

We consider the empirical investment equation that generalizes equation (6) as

(7) $$ {y}_{it}={\alpha}_i+\beta {q}_{it}+{Z}_{it}\theta +{\omega}_{it}+{u}_{it}, $$

where $ {y}_{it} $ is the investment ratio and $ {\alpha}_i $ is the firm fixed effect. Compared to the investment equation (6), without loss of generality, we normalize the coefficient on $ {\omega}_{it} $ to be 1 because this persistent shock is an unobserved factor.

The true $ q $ may not be directly observable and can only be measured with error as $ {q}_{it}={q}_{it}^{\ast }+{e}_{it} $ , where $ {q}_{it}^{\ast } $ and $ {e}_{it} $ denote the true $ q $ and possible measurement error, respectively. The vector of state variables, $ {Z}_{it} $ , includes other potential observable factors of investment, such as cash flow (or leverage) and firm size, which proxy for firm heterogeneity and demand shocks. These variables can be incorporated into the investment equation, and their inclusion in the estimation does not introduce additional endogeneity problems once the unobserved $ {\omega}_{it} $ is controlled for. However, if $ {\omega}_{it} $ is omitted, these additional observed factors, including cash flow, can become endogenous regressors as well. This highlights the importance of controlling for the unobserved persistent shock in the investment equation to consistently estimate coefficients of both q and other observed factors.

The investment equation contains two unobserved shocks, $ {\omega}_{it} $ and $ {u}_{it} $ . Motivated by equation (5), here, we allow $ {q}_{it} $ to be correlated with $ {\omega}_{it} $ . Both $ {q}_{it} $ and $ {Z}_{it} $ are not correlated with the exogenous shock $ {u}_{it} $ . Following a standard setting in the literature to deal with the persistent error (Olley and Pakes (Reference Olley and Pakes1996), Blundell and Bond (Reference Blundell and Bond2000)), we assume $ {\omega}_{it} $ follows a Markov process, such as a simple autoregressive process of order 1 (AR(1)).Footnote ⁴

Assumption III.1 Let $ {\omega}_{it} $ be an unobserved persistent shock, a factor of the investment cost in equation (1). We assume that

(8) $$ {\omega}_{it}={\rho \omega}_{i,t-1}+{\xi}_{it}, $$

where $ {\xi}_{it} $ denotes the innovation term in the process, and the dynamic parameter $ \rho $ satisfies $ \left|\rho \right|<1 $ .

The investment equation can be estimated with or without the firm fixed effect $ {\alpha}_i $ . We primarily focus on the case with the fixed effect in our approach; the estimation without the fixed effect can proceed without the first-order differences to remove the fixed effect, as in Section III.B below. For the empirical implementation of the estimator, in Section V, we provide more details on the model with and/or without the firm fixed effect.

1. Estimation without Measurement Error

We first consider the model where $ {q}_{it} $ is measured without error and we set $ {q}_{it}={q}_{it}^{\ast } $ where $ {q}_{it}^{\ast } $ denotes the true $ q $ . Applying generalized differencing to equation (7), we obtain

$$ {y}_{it}=\left(1-\rho \right){\alpha}_i+\rho {y}_{i,t-1}+\beta \left({q}_{it}-\rho {q}_{i,t-1}\right)+\left({Z}_{it}-\rho {Z}_{i,t-1}\right)\theta +{u}_{i,t}-\rho {u}_{i,t-1}+{\xi}_{it}. $$

By taking the first-order differences to remove the firm fixed effect, we then obtain

$$ \Delta {y}_{it}=\rho \Delta {y}_{i,t-1}+\beta \left(\Delta {q}_{it}-\rho \Delta {q}_{i,t-1}\right)+\left(\Delta {Z}_{it}-\rho \Delta {Z}_{i,t-1}\right)\theta +\Delta {u}_{it}-\rho \Delta {u}_{i,t-1}+\Delta {\xi}_{it}. $$

We now utilize a set of timing and information set assumptions as our identifying conditions. We make the following assumptions.

Assumption III.2 Let $ {u}_{it} $ be an idiosyncratic shock in the investment equation (7) and $ {\xi}_{it} $ be the innovation term in the persistent shock process (8). Let $ {Z}_{it} $ denote other observed factors of investment, which are conditionally mean independent of the innovation $ {\xi}_{it} $ . The shocks satisfy

$$ E\left[{u}_{it}|{J}_{it}\right]=0,E\left[{u}_{i{t}^{\prime }}|{Z}_{it}\right]=0 $$

for all $ t $ and $ {t}^{\prime } $ and

$$ E\left[{\xi}_{it}|{J}_{i,t-1},{Z}_{it}\right]=0,\mathrm{or}\;E\left[\Delta {\xi}_{it}|{J}_{i,t-2},\Delta {Z}_{it},\Delta {Z}_{i,t-1}\right]=0 $$

where $ {J}_{it} $ denotes the information available to the firm i at a point in time $ t $ when the firm makes the investment decision.

Note that, by construction, $ {J}_{it} $ includes all current observables at the time of the investment decision and their lags. For example, $ {J}_{it} $ may include $ {q}_{it} $ , $ {Z}_{it} $ , and $ {y}_{i,t-1} $ (and their respective lags). However, for estimation, the valid instruments may consist of only a subset of $ {J}_{it} $ or may include additional available variables, depending on the moment conditions, as detailed in our data-driven IV selection.

Assumption III.2 states that i) $ {u}_{it} $ , the exogenous shock to adjustment cost, is not systematically over or underpredicted, given the information available at time t, and this shock is also strictly exogenous with respect to $ {Z}_{it} $ . It also imposes that ii) the innovation of the persistent shock process is uncorrelated with any information available at time $ t-1 $ or other observed factors $ {Z}_{it} $ ; this is reasonable since $ {\omega}_{it} $ follows an exogenous Markov process. Note that this assumption allows other observed factors $ {Z}_{it} $ to be correlated with $ {\omega}_{it} $ but not with the innovation term $ {\xi}_{it} $ .

In the dynamic panel literature, Assumption III.2 is often referred to as $ {J}_{i,t-1} $ including “predetermined” variables. Our identifying conditions are also motivated by rational expectations in the sense that, given available information, a firm does not over or underinvest on average. In other words, from the available information, we cannot predict systematic over or underinvestment by firms. Ackerberg (Reference Ackerberg2023) further provides details on how these assumptions can be strengthened or relaxed. An important point he elaborates is that what matters is not only the timing of when firms choose the “predetermined” variables but also what they know at that time. In this sense, these restrictions are referred to as the timing and information set assumptions, not only timing assumptions.

These assumptions allow that $ {q}_{it} $ is potentially endogenous, even being free of measurement error. Under Assumptions III.1 and III.2, we then obtain the moment condition

$$ E\left[\Delta {u}_{it}-\rho \Delta {u}_{i,t-1}+\Delta {\xi}_{it}|{J}_{i,t-2},\Delta {Z}_{it},\Delta {Z}_{i,t-1}\right]=0, $$

from which we obtain the moment condition for GMM estimation

(9) $$ E\left[\Delta {y}_{it}-\left\{\rho \Delta {y}_{i,t-1}+\beta \left(\Delta {q}_{it}-\rho \Delta {q}_{i,t-1}\right)+\left(\Delta {Z}_{it}-\rho \Delta {Z}_{i,t-1}\right)\theta \right\}|{J}_{i,t-2},\Delta {Z}_{it},\Delta {Z}_{i,t-1}\right]=0. $$

2. Estimation with Measurement Error

Next, we consider the measurement error in $ {q}_{it}={q}_{it}^{\ast }+{e}_{it} $ . The regression equation derived from equation (7) becomes

$$ {\displaystyle \begin{array}{c}\Delta {y}_{it}=\rho \Delta {y}_{i,t-1}+\beta \left(\Delta {q}_{it}-\rho \Delta {q}_{i,t-1}\right)+\left(\Delta {Z}_{it}-\rho \Delta {Z}_{i,t-1}\right)\theta \\ {}+\Delta {u}_{it}-\rho \Delta {u}_{i,t-1}+\Delta {\xi}_{it}-\beta \left(\Delta {e}_{it}-\rho \Delta {e}_{i,t-1}\right).\end{array}} $$

We further assume that the measurement error is not persistent in the sense that it is not correlated with lagged information $ {J}_{i,t-1} $ , and is also not correlated with other observable factors of the investment $ {Z}_{it} $ , as follows:

Assumption III.3 Let $ {q}_{it}={q}_{it}^{\ast }+{e}_{it} $ , where $ {q}_{it}^{\ast } $ denotes the true q and $ {e}_{it} $ denotes its measurement error. The measurement error satisfies for all $ t $ and $ {t}^{\prime } $ ,

$$ E\left[{e}_{it}|{J}_{i,t-1}\right]=0\hskip1em \mathrm{and}\hskip1.12em E\left[{e}_{i{t}^{\prime }}|{Z}_{it}\right]=0. $$

This assumption about the measurement error commonly appears in the literature, which rules out $ q $ being systematically mismeasured. This is a reasonable condition since the market’s perception of the firm’s true q is continuously updated by rationally incorporating information available at the market up to the current date. The assumption of the measurement error being uncorrelated with the first-order lagged information, $ {J}_{i,t-1} $ , is plausible, given the annual frequency of the data that is empirically used to estimate the investment equation. For instance, if the measurement error follows a moving average process of order 1 (MA(1)), the assumption is satisfied. It follows from this assumption that

$$ E\left[\Delta {e}_{it}|{J}_{i,t-2},{Z}_{it}\right]=0\hskip1.12em \mathrm{and}\hskip1.12em E\left[\Delta {e}_{i,t-1}|{J}_{i,t-3},{Z}_{it}\right]=0. $$

By combining these conditional moment conditions, we then obtain

$$ E\left[\Delta {e}_{it}-\rho \Delta {e}_{i,t-1}|{J}_{i,t-3},\Delta {Z}_{it},\Delta {Z}_{i,t-1}\right]=0. $$

From this result, it is clear that, given Assumption III.3, the moment condition (9) can be made robust to the measurement error of $ {q}_{it} $ by using the following moment condition:Footnote ⁵

(10) $$ E\left[\Delta {y}_{it}-\left\{\rho \Delta {y}_{i,t-1}+\beta \left(\Delta {q}_{it}-\rho \Delta {q}_{i,t-1}\right)+\left(\Delta {Z}_{it}-\rho \Delta {Z}_{i,t-1}\right)\theta \right\}|{J}_{i,t-3},\Delta {Z}_{it},\Delta {Z}_{i,t-1}\right]=0. $$

It is worth mentioning that most studies assume the classical measurement error in $ q $ and do not allow for a persistent measurement error. Nevertheless, Assumption III.3 can be modified to allow for a more persistent measurement error; this would require changing the conditioning variables in the moment condition. In our empirical applications (Section V), we examine this scenario using a more persistent process and find that the measurement error process outlined in Assumption III.3 aligns more appropriately with the empirical settings.

With the use of additional notation, we can also allow the coefficients $ \beta $ and $ \gamma $ to vary by time or period. In Section V, we provide further details on how to choose instruments to implement GMM estimation, based on these timing and information set assumptions.

B. Implementation of Estimation

We discuss here how to implement the GMM estimation for the investment equation (7), with measurement error in the measured $ {q}_{it}={q}_{it}^{\ast }+{e}_{it} $ , where $ {q}_{it}^{\ast } $ denotes the true q. We consider two cases: the model without and with the firm fixed effect, respectively:

$$ {\iota}_{it}=\alpha +\beta {q}_{it}+{Z}_{it}\theta +{\omega}_{it}+{u}_{it}-\beta {e}_{it}, $$

and

$$ {\iota}_{it}={\alpha}_i+\beta {q}_{it}+{Z}_{it}\theta +{\omega}_{it}+{u}_{it}-\beta {e}_{it}, $$

where we now use the notation $ {\iota}_{it} $ , instead of $ {y}_{it} $ , to denote various investment measures in our analyses.Footnote ⁶ Here $ {\alpha}_i $ denotes the firm fixed effect; $ {\omega}_{it} $ denotes the unobserved persistent shock; $ {u}_{it} $ is an exogenous shock to the adjustment cost; $ {e}_{it} $ is the measurement error in $ {q}_{it}^{\ast } $ .

The model without the firm fixed effect, after applying the generalized differencing due to the AR(1) process of the persistent shock (8), yields

$$ {\displaystyle \begin{array}{c}{\iota}_{it}=\alpha \left(1-\rho \right)+{\rho \iota}_{i,t-1}+\beta \left({q}_{it}-\rho {q}_{i,t-1}\right)+\left({Z}_{it}-\rho {Z}_{i,t-1}\right)\theta \\ {}+{u}_{it}-\rho {u}_{i,t-1}+{\xi}_{it}-\beta \left({e}_{it}-\rho {e}_{i,t-1}\right).\end{array}} $$

We note that the variables $ \left\{{Z}_{it},{Z}_{i,t-1}\right\} $ satisfy the moment condition and serve as instruments for themselves, while the variables $ \left\{{\iota}_{i,t-1},{q}_{it},{q}_{i,t-1}\right\} $ do not. Therefore, we can use the following set of further lagged variables as excluded instrumental variables (IVs), because they are not correlated with the error terms $ \left[{u}_{it}-\rho {u}_{i,t-1}+{\xi}_{it}-\beta \left({e}_{it}-\rho {e}_{i,t-1}\right)\right]: $

$$ \left\{\begin{array}{c}{\iota}_{i,t-2},{\iota}_{i,t-3},{\iota}_{i,t-4},\dots \\ {}{q}_{i,t-2},{q}_{i,t-3},{q}_{i,t-4},\dots \\ {}{Z}_{i,t-2},{Z}_{i,t-3},{Z}_{i,t-4},\dots \end{array}\right\} $$

We justify these IVs based on the assumptions about timing and information set, as discussed in the previous section, and we adopt data-driven criteria to select IVs among this set of variables, as we detail in Section III.C.

Define the vector of parameters $ \vartheta \equiv {\left(\alpha, \beta, {\theta}^{\prime },\rho \right)}^{\prime } $ . Let $ {H}_{it} $ be a $ K\times 1 $ vector that stacks the IVs we select, given firm i and time t. Let $ \hat{W} $ be a $ K\times K $ weighting matrix satisfying the $ \hat{W}{\to}_pW $ condition, with a symmetric positive definite matrix W. The GMM estimator of $ \vartheta $ is defined as

(11) $$ \hat{\vartheta}\equiv {\mathrm{argmin}}_{\theta}\;{g}_n{\left(\vartheta \right)}^{\prime}\hat{W}{g}_n\left(\vartheta \right), $$

with

$$ {g}_n\left(\vartheta \right)\equiv \frac{1}{n}\sum \limits_{i=1}^n\sum \limits_{t={t}_0}^{T_i}{H}_{it}\cdot \left({\iota}_{it}-\alpha \left(1-\rho \right)-{\rho \iota}_{i,t-1}-\beta \left({q}_{it}-\rho {q}_{i,t-1}\right)-\left({Z}_{it}-\rho {Z}_{i,t-1}\right)\theta \right), $$

where $ {t}_0 $ is determined by the availability of lagged variables in the instruments $ {H}_{it} $ , depending on the choice of lags in the instruments. Under standard regularity conditions for GMM, the estimator achieves consistency and asymptotic normality:

$$ \sqrt{n}\left(\hat{\vartheta}-\vartheta \right){\to}_dN\left(0,{V}_{\vartheta}\right), $$

with the asymptotic variance–covariance matrix $ {V}_{\vartheta } $ .

Similarly, for the model with the firm fixed effect, after applying the generalized differencing to the first-differenced equation, we obtain

$$ {\displaystyle \begin{array}{c}\Delta {\iota}_{it}=\rho \Delta {\iota}_{i,t-1}+\beta \left(\Delta {q}_{it}-\rho \Delta {q}_{i,t-1}\right)+\left(\Delta {Z}_{it}-\rho \Delta {Z}_{i,t-1}\right)\theta \\ {}+\Delta {u}_{it}-\rho \Delta {u}_{i,t-1}+\Delta {\xi}_{it}-\beta \left(\Delta {e}_{it}-\rho \Delta {e}_{i,t-1}\right).\end{array}} $$

In this case, the variables $ \left\{\Delta {Z}_{it},\Delta {Z}_{i,t-1}\right\} $ satisfy the moment condition, while the variables $ \left\{\Delta {\iota}_{i,t-1},\Delta {q}_{it},\Delta {q}_{i,t-1}\right\} $ do not. Then the following set of further lagged variables can be used as excluded IVs, because they are orthogonal to the error terms $ \left[\Delta {u}_{it}-\rho \Delta {u}_{i,t-1}+\Delta {\xi}_{it}-\beta \left(\Delta {e}_{it}-\rho \Delta {e}_{i,t-1}\right)\right] $ :

(12) $$ \left\{\begin{array}{l}{\iota}_{i,t-3},{\iota}_{i,t-4},{\iota}_{i,t-5},\dots \\ {}{q}_{i,t-3},{q}_{i,t-4},{q}_{i,t-5},\dots \\ {}{Z}_{i,t-3},{Z}_{i,t-4},{Z}_{i,t-5},\dots \end{array}\right\}. $$

We note that the set of instruments consisting of the differenced version of the IVs can also be used in place of the IVs above:

(13) $$ \left\{\begin{array}{l}\Delta {\iota}_{i,t-3},\Delta {\iota}_{i,t-4},\Delta {\iota}_{i,t-5},\dots \\ {}\Delta {q}_{i,t-3},\Delta {q}_{i,t-4},\Delta {q}_{i,t-5},\dots \\ {}\Delta {Z}_{i,t-3},\Delta {Z}_{i,t-4},\Delta {Z}_{i,t-5},\dots \end{array}\right\}. $$

Define the vector of parameters $ \tilde{\vartheta}\equiv {\left(\beta, {\theta}^{\prime },\rho \right)}^{\prime } $ . Then, the GMM estimator of $ \tilde{\vartheta} $ is defined as in equation (11):

(14) $$ \hat{\tilde{\vartheta}}\equiv {\mathrm{argmin}}_{\tilde{\vartheta}}\;{\tilde{g}}_n{\left(\tilde{\vartheta}\right)}^{\prime}\hat{W}{\tilde{g}}_n\left(\tilde{\vartheta}\right), $$

with

$$ {\tilde{g}}_n\left(\tilde{\vartheta}\right)\equiv \frac{1}{n}\sum \limits_{i=1}^n\sum \limits_{t={t}_1}^{T_i}{\tilde{H}}_{it}\cdot \left(\Delta {\iota}_{it}-\rho \Delta {\iota}_{i,t-1}-\beta \left(\Delta {q}_{it}-\rho \Delta {q}_{i,t-1}\right)-\left(\Delta {Z}_{it}-\rho \Delta {Z}_{i,t-1}\right)\theta \right). $$

$ {\tilde{H}}_{it} $ denotes the vector that stacks the instruments equation (12) or equation (13), and $ {t}_1 $ is determined by the availability of lagged variables in the instruments $ {\tilde{H}}_{it} $ , depending on the choice of lags in the instruments. We discuss our criteria for selecting instruments and provide some practical guidelines in Section III.C.

In practice, the proposed estimator is easy to implement in standard computing software. For illustrative purposes, we utilize the Stata command, gmm, to implement the proposed estimator in the empirical estimation. We use the robust option for the weighting matrix and cluster the standard errors of the parameter estimates at the firm level.

C. Selection of Instrumental Variables

We adopt data-driven criteria to select IVs that satisfy legitimate instrumental variable conditions. Since the number of IVs can be more than the number of endogenous variables as long as the moment condition is satisfied, in principle, the model can be overidentified. So, our first criterion is Hansen’s J-test for overidentification. To ensure consistency of the estimator and its desirable finite sample performance, IVs should not suffer from weak instrument problems. The second criterion is strong instrument tests. In particular, we consider Sanderson and Windmeijer’s F-tests for weak identification and underidentification since multiple endogenous variables exist at the moment condition for the investment equation. Third, we select the specification that minimizes the residual of the GMM objective function (11) or (14) as small as possible. This guarantees that the estimates are the global minimizers of the optimization problem. Lastly, we conduct AR(1) and AR(2) diagnostic tests on the regression residuals obtained from our estimation (using, e.g., Stata’s arima command). These tests examine whether the residuals exhibit first- or second-order serial correlation. The absence of significant higher-order autocorrelation provides additional support that our moment conditions are not misspecified and that the GMM framework is appropriately designed. These practical criteria guarantee that the selected instruments are valid and relevant for the moment conditions.

A. Motivating Preliminary Analyses

We first estimate the investment equation using a higher-order polynomial OLS model, where a nonlinear function of the state variables, such as cash flow (CF), firm size (lnK), and employment (lnN_K), is considered as a benchmark drawn from the recent literature. In this literature, Gala et al. (Reference Gala, Gomes and Liu2020) argue that observed state variables explain corporate investment better than q and propose a flexible polynomial regression approach to avoid model misspecification, while Song and Wee (Reference Song and Wee2026) document heterogeneity in the investment-q relation, using nonlinear investment equations. The results are reported in Appendix B.

We start by estimating a simple model with polynomial approximation in the state variables and confirm that the state variables in place of q explain the investment. However, we find that q remains significant in the estimation equations even after controlling for higher-order polynomials of these variables and firm fixed effects. We then add the estimated TFP to a linear investment equation and find that TFP is statistically significant at the 1% level. Estimating a more flexible model with higher-order terms does not alter the result. These estimation results motivate us to treat $ q $ and the unobserved persistent shock as the main factor of the investment function. Since the coefficients of the higher-order terms have small magnitudes and little economic significance, our main analyses primarily focus on the linear model with a firm fixed effect, while subperiod analysis and nonlinear models serve as robustness checks.

We next formally implement our GMM estimation approach to account for measurement error in q and the potential unobserved persistent shock to investment policy. The results are presented in Table 2. We estimate our model both with and without TFP included as an additional observed state variable to assess whether our approach can effectively account for TFP when it is unobserved.

TABLE 2Preliminary GMM Regression Results Using TFP Proxy

TABLE 2 Long description

The table consists of four columns: Variables, 1, 2, and 3.

Row 1: x. Column 1 is 0.009*** (3.15). Column 2 is 0.016*** (14.29). Column 3 is 0.025*** (16.02).

Row 2: C F. Column 1 is 0.006** (2.12). Column 2 is 0.003 (1.44). Column 3 is minus 0.001 (minus 0.24).

Row 3: ln K (size). Column 1 is minus 0.123*** (minus 26.56). Column 2 is minus 0.128*** (minus 22.63). Column 3 is minus 0.157*** (minus 17.55).

Row 4: ln N sub K (employment). Column 1 is 0.066*** (13.58). Column 2 is 0.048*** (11.21). Column 3 is 0.048*** (10.16).

Row 5: omega. Column 1 is blank. Column 2 is 0.160*** (4.55). Column 3 is 0.435*** (10.79).

Row 6: T F P. Column 1 is 0.016*** (5.78). Column 2 is blank. Column 3 is 0.004 (1.37).

Row 7: Firm-year sample. All columns are 107,183.

Row 8: e(Q). Column 1 is 0.00000. Column 2 is 0.00294. Column 3 is 0.00279.

Row 9: P-value of Hansen J. Column 1 is 0.78. Column 2 is 0.24. Column 3 is 0.55.

Row 10: F stats of inv(t minus 1). Column 1 is blank. Column 2 is 160.58. Column 3 is 33.55.

Row 11: F stats of x(t minus 1). Column 1 is 27.72. Column 2 is 40.30. Column 3 is 38.18.

Row 12: F stats of x(t minus 2). Column 1 is blank. Column 2 is 69.65. Column 3 is 67.78.

Row 13: S W P of inv(t minus 1). Column 1 is blank. Column 2 is 0.00. Column 3 is 0.00.

Row 14: S W P of x(t minus 1). Column 1 is 0.00. Column 2 is 0.00. Column 3 is 0.00.

Row 15: S W P of x(t minus 2). Column 1 is blank. Column 2 is 0.00. Column 3 is 0.00.

Row 16: S W kappa-squared P of inv(t minus 1). Column 1 is blank. Column 2 is 0.00. Column 3 is 0.00.

Row 17: S W kappa-squared P of x(t minus 1). Column 1 is 0.00. Column 2 is 0.00. Column 3 is 0.00.

Row 18: S W kappa-squared P of x(t minus 2). Column 1 is blank. Column 2 is 0.00. Column 3 is 0.00.

Row 19: Firm F E. All columns are Yes.

As reported in column 1, we estimate the GMM regression with TFP, q, cash flow, size, employment, and a firm fixed effect in the investment function, while we do not account for the unobserved shock $ \omega $ . We find that both q and TFP are statistically significant for this specification. The coefficient of q is 0.009 and significant at the 1% level, and the coefficient of TFP is 0.016 and also significant at the 1% level. In column 2, we account for the unobserved shock $ \omega $ using our approach instead of the TFP proxy. However, we estimate the model using firms for which TFP is available to facilitate comparison. The result shows that q remains significant at a 1% level with a magnitude of 0.016, which is improved from 0.009 in column 1 when we include TFP as an additional state variable.

Next, in column 3, we include TFP as a control variable and simultaneously account for $ \omega $ in the GMM estimation. Here, interestingly, TFP becomes statistically insignificant, while the AR(1) parameter of $ \omega $ remains significant at the 1% level. The coefficient of q is statistically significant at the 1% level with a magnitude of 0.025. Notably, TFP is redundant in this model specification. This indicates that accounting for the persistent unobserved shock using our approach can effectively subsume this TFP shock, which is not fully captured by the observed $ q $ , even if it is omitted.

Overall, the results in Table 2 highlight the advantages of accounting for the persistent shock in estimating the investment equation, thereby supporting the efficacy of our proposed approach.Footnote ⁷

B. Main Results

We now consider the investment equations for the full sample period from 1975 to 2021, for which we include $ q $ , along with other state variables, in the investment function and account for the unobserved persistent shock using our proposed approach. The estimation results are reported in Table 3 (Panel A).

TABLE 3Main Results with Full Sample Using the Proposed Approach

TABLE 3 Long description

Panel A. Regression Results. This panel compares one O L S model (column 1) with six G M M models (columns 2 through 6).

- The first row shows coefficients for x, which are all positive and statistically significant at the 1 percent level, ranging from 0.006 in O L S to 0.024 in G M M column 3.

- The second row shows C F coefficients, which are generally small and positive.

- The third and fourth rows show ln K (size) and ln N underscore K (employment). Size has a negative correlation (around -0.13 to -0.18) while employment has a positive correlation (0.03 to 0.05).

- The fifth row shows omega coefficients for models 3 through 6, ranging from 0.159 to 0.503.

- Diagnostic rows include Measurement error (No, M A (1), or M A (2)), Firm-year sample (149,429 for all), R-squared (0.209 for O L S), and various F-statistics and S W P values for instruments at t minus 1 and t minus 2.

Panel B. Coefficient Comparison. This panel compares the coefficient and standard error of x across four models: 1 O L S, 2 G M M-M A (1), 4 G M M-M A (1) and omega, and 5 G M M-omega.

- The coefficient of x increases from 0.006 in O L S to 0.017 in G M M-M A (1) and omega.

- Standard errors are lowest in O L S (0.00022) and highest in G M M-M A (1) (0.00221).

- The bottom section lists t-statistics for the difference of x between models: O L S versus G M M-M A (1) is 1.35; G M M M A (1) versus G M M M A (1) and omega is 3.14; O L S versus G M M M A (1) and omega is 8.60; and G M M-omega versus G M M-M A (1) and omega is 2.57.

We first estimate the OLS regression with q, cash flow, size, and employment in column 1. The results show that when the OLS does not account for the persistent shock and the measurement error in q, the coefficient of q is significant at the 1% level, but its magnitude is as small as 0.006, suggesting a downward bias. In columns 2–6, we implement our proposed approach in different model specifications to examine the robustness of the results. Columns 2 and 4 are based on the specifications of the measurement error as in Assumption III.3 (e.g., MA(1)). Column 5 assumes no measurement error in q and column 6 assumes more persistent measurement error in q (e.g., MA(2)). We use the criteria for choosing IVs discussed in the previous section and only report the estimation results that satisfy the selection criteria to avoid redundant tables. Throughout the section, the employed IVs are reported at the top of each table. The residual (e(Q)) represents the difference between the actual observed values of the dependent variable and those predicted by the model. In the context of GMM regression, a low residual means that the predicted values of investment generated by the GMM estimates are close to their observed values. The P-value of Hansen’s J-test (P-value of Hansen J) is reported to check the validity of the IVs. SW P-value (SWP) is Sanderson and Windmeijer’s F-test for the weak identification problem. SW $ {\kappa}^2 $ P-value (SW $ {\kappa}^2 $ P) is Sanderson and Windmeijer’s F-test for the underidentification problem. In addition, we check diagnostic tests of the autoregressive process on the estimation residuals and report them in Appendix C. These test results confirm that the selected IVs are valid and strong.Footnote ⁸

We start with the MA(1) measurement error specification in columns 2–4. Column 2 includes q, cash flow, and size and employment as other state variables. Column 3 eliminates size and employment and only includes q, cash flow, and the unobserved persistent shock $ \omega $ . First, the result in column 2 confirms that the state variables are significant in explaining corporate investment. Furthermore, we find both q and the dynamic parameter (AR(1)) of $ \omega $ are statistically significant at the 1% level. When we include the unobserved persistent shock in column 3, it is highly significant at a 1% level, suggesting that it is a significant factor in the investment function. In column 4, we include the state variables in addition to the unobserved persistent shock. We can infer that the unobserved persistent shock is still a significant factor in the investment function, and q remains statistically and economically significant after controlling for cash flow, size, and employment. We also observe an increased magnitude of the coefficient of q from 0.006 in column 1, the OLS estimation, to 0.017 in column 4, the GMM estimation. This discrepancy arises from both attenuation bias and omitted variable bias inherent in OLS estimation. The magnitude of the coefficient of q converted to dollar value is $1.72 million increase in capital expenditures with a one-unit increase in q for the median firms in the sample. For the top and bottom quartile firms in the sample, the values are $9.67 million and $0.44 million.Footnote ⁹ We also compare our estimates with those from Erickson and Whited (Reference Erickson and Whited2000) and Peters and Taylor (Reference Peters and Taylor2017) in Appendix D. Overall, the evidence highlights the advantage of our empirical approach, which accounts for both the measurement error in q and the unobserved persistent shock to investment.

Next, to gauge the empirical relevance of the measurement error in q and the unobserved persistent shock in our estimation separately, we examine the specification that does not account for either the measurement error or the persistent shock. The results are reported in columns 4–5. First, under the specification without measurement error in column 5, the persistent shock is statistically significant, and the coefficient of q is significantly larger than the one in column 1. This implies that the OLS estimator is downward-biased if the persistent shock is omitted. Second, comparing the results in column 4 under the MA(1) specification with those in column 5 without measurement error, we find that measurement error in q attenuates its estimated coefficient. After accounting for measurement error, the coefficient of q in column 4 is significantly larger than that in column 5. Specifically, the difference in the coefficients of q is statistically significant, with a t-statistic of 2.57.

We also investigate the case where the measurement error of q is more persistent, such as an MA(2) process, and report the results in column 6. In this case, we modify the conditioning variables in the moment condition from $ {J}_{i,t-1} $ to $ {J}_{i,t-2} $ . Under the MA(2) specification, the coefficients on q and the persistent shock remain very close to those under the MA(1) specification in column 4, and the differences are not statistically significant, indicating that our results are robust to allowing for more persistent measurement error. Specifically, the t-statistics of the coefficient differences are 0.219 for q and 0.979 for $ \omega $ , respectively. Thus, the more parsimonious model in column 4 can be selected. As a result, the estimation confirms that the investment equation, which includes the persistent shock $ \omega $ and measurement error (column 4), is the most preferred model specification. In particular, these results highlight the importance and empirical relevance of addressing both the omitted persistent shock and measurement error.

C. Estimation with Leverage

Existing studies, such as Whited (Reference Whited1992) and Hennessy, Levy, and Whited (Reference Hennessy, Levy and Whited2007), suggest that financial liabilities should be a state variable for the optimal investment policy. In this section, we use leverage as a relevant state variable. We employ two variables: leverage (the sum of long-term and short-term debt, scaled by total assets) and net leverage (the total debt netting cash and short-term investments, scaled by total assets). We estimate both GMM and OLS models by including either leverage or net leverage in place of cash flow. The results are presented in Panels A and B of Table 4, respectively.

TABLE 4Estimation with Leverage

TABLE 4 Long description

The table is divided into two main sections: Panel A for G M M Estimation and Panel B for O L S Estimation. Both panels use four numbered columns (1 through 4) representing different model specifications.

Panel A: G M M Estimation with Leverage

- Variable q: Coefficients range from 0.037 to 0.040, all significant at the 1 percent level, with t-statistics between 13.31 and 16.06.

- Leverage: Negative coefficients of -0.102 (Model 1) and -0.109 (Model 3).

- Net leverage: Negative coefficients of -0.084 (Model 2) and -0.097 (Model 4).

- ln K (size): Negative coefficients ranging from -0.088 to -0.119.

- ln N over K (employment): Positive coefficients from 0.012 to 0.020.

- ln Y over K (sales): Positive coefficients of 0.007 in Models 3 and 4.

- Omega: Coefficients range from 0.111 to 0.468.

- Diagnostics: Firm-year sample sizes are approximately 149,000 for Models 1-2 and 131,000 for Models 3-4. P-values of Hansen J range from 0.19 to 0.84. F-statistics and S W P values for various lags are provided, with all S W P values at 0.00.

- Firm F E: Included (Yes) for all models.

Panel B: O L S Estimation with Leverage

- Variable q: Coefficients are consistently 0.005 or 0.006, significant at the 1 percent level.

- Leverage: Negative coefficients of -0.140 (Model 1) and -0.118 (Model 3).

- Net leverage: Negative coefficients of -0.151 (Model 2) and -0.130 (Model 4).

- ln K (size): Negative coefficients ranging from -0.158 to -0.178.

- ln N over K (employment): Positive coefficients from 0.042 to 0.056.

- ln Y over K (sales): Positive coefficients of 0.027 and 0.028 in Models 3 and 4.

- R-squared: Values range from 0.197 to 0.227.

- Firm F E: Included (Yes) for all models.

In Panel A, columns 1 and 2, we find that q is statistically significant in the GMM estimations, while leverage and net leverage are negatively associated with optimal investment. In columns 3 and 4, we include the sales-to-capital ratio, but the results remain robust. The negative coefficients on leverage and net leverage are consistent with the findings in Gala et al. (Reference Gala, Gomes and Liu2020) that higher financial liabilities constrain investment. Panel B presents the OLS results. Comparing the magnitude of the coefficients on q, the GMM estimates (ranging from 0.037 to 0.040 in Panel A) are substantially larger than the OLS estimates (ranging from 0.005 to 0.006 in Panel B), indicating that controlling for endogeneity strengthens the sensitivity of investment to q.

D. Subperiod Analysis

To examine possible structural change due to the financial crisis in the investment equation over time, we split the sample into two subperiods (1975–2009 and 2010–2021) and estimate the equations for each subperiod. The OLS and GMM estimation results are reported in Table 5. As before, we estimate investment equations with a firm fixed effect. The OLS results show that the coefficients of q are statistically significant in both subperiods, with the magnitude of the coefficient being smaller for the subperiod of 2010–2021. The magnitude of q for the estimation before 2010 is 0.007, and decreases to 0.004 for the estimation period of 2010 to 2021. The GMM results in columns 3 and 4 show that q and $ \omega $ are significant in both subperiods. The coefficient on $ \omega $ increases from 0.122 (for the period before 2010) to 0.238 (after 2010). Thus, accounting for the unobserved persistent shock has become increasingly important in the estimation of investment equations in recent periods. We can also infer that q has exhibited increasing importance since 2010, in contrast to the results from the OLS. It is worth noting that in both subperiods, the magnitudes of the coefficient of q in the GMM estimation are larger than those of the OLS estimation. It is consistent with the main results that the investment-q sensitivity becomes larger after controlling for measurement error and the unobserved persistent shock using our approach.

TABLE 5Subperiod Analyses

TABLE 5 Long description

The table is organized into four main columns under two primary estimation methods: O L S and G M M. Each method is split into two sub-periods: <2010 and 2010–2021.

Key variables and their coefficients (with t-statistics in parentheses) include:

- x: O L S values are 0.007 (24.94) and 0.004 (13.27); G M M values are 0.013 (12.82) and 0.017 (6.81).

- C F: O L S values are 0.014 (6.42) and 0.002 (0.71); G M M values are 0.010 (3.50) and -0.005 (-1.51).

- ln K (size): O L S values are -0.195 (-47.58) and -0.158 (-27.26); G M M values are -0.146 (-23.19) and -0.133 (-14.44).

- ln N_K (employment): O L S values are 0.062 (13.55) and 0.036 (7.52); G M M values are 0.044 (8.40) and 0.020 (3.61).

- omega: Only reported for G M M with values 0.122 (3.25) and 0.238 (4.30).

Summary statistics:

- Firm-year sample: 108,862 for <2010 and 40,567 for 2010–2021.

- R-squared (O L S only): 0.224 and 0.179.

- P-value of Hansen J (G M M only): 0.18 and 0.85.

- F stats and S W P values for various instruments are provided for the G M M columns, showing high significance (0.00 to 0.04).

- Firm F E: Yes for all models.

E. Estimation of Other Investment Types

The results presented so far are for standard investment, defined as capital expenditures scaled by physical capital. In this section, we examine the robustness of our GMM estimation approach to different types of investment, that is, total, physical, and intangible investments. We follow Peters and Taylor (Reference Peters and Taylor2017) to construct these variables.Footnote ¹⁰ The results are reported in Table 6. Panels A and B present GMM and OLS estimation results, respectively. In the GMM estimations in columns 1–3 for total, physical, and intangible investment, respectively, we find that total q is statistically significant at the 1% level across different types of investment, after controlling for the state variables of cash flow, size, and employment. In fact, the total $ q $ includes both physical and intangible capital. The unobserved persistent shock $ \omega $ remains a significant factor in the investment-q estimation. This reinforces our argument that $ \omega $ may capture other unobserved investment factors, such as technology or information efficiency shocks, and is still important to explain investment even after controlling for observed intangible measures proposed in the existing literature.

TABLE 6Total, Physical, and Intangible Investment

TABLE 6 Long description

The table is divided into two sections: Panel A for G M M Estimations and Panel B for O L S Estimations. Both panels use three columns: 1 Total, 2 Physical, and 3 Intangible.

Panel A (G M M Estimations):

* q_total: Total 0.091, Physical 0.060, Intangible 0.034. All are significant at the 1 percent level.

* C F_total: Total 0.031, Physical 0.004, Intangible 0.023.

* ln K (size): Total minus 0.125, Physical minus 0.075, Intangible minus 0.046.

* ln N_K (employment): Total minus 0.001, Physical 0.005, Intangible minus 0.008.

* Omega: Total 0.504, Physical 0.532, Intangible 0.597.

* Firm-year sample: 149,429 for all columns.

* P-value of Hansen J: Total 0.80, Physical 0.47, Intangible 0.59.

* F stats of q(t minus 1): Total 88.08, Physical 158.52, Intangible 96.21.

* Firm F E: Yes for all columns.

Panel B (O L S Estimations):

* q_total: Total 0.028, Physical 0.015, Intangible 0.011.

* C F_total: Total 0.105, Physical 0.058, Intangible 0.046.

* ln K (size): Total minus 0.126, Physical minus 0.081, Intangible minus 0.037.

* ln N_K (employment): Total 0.024, Physical 0.017, Intangible 0.008.

* Adjusted R-squared: Total 0.249, Physical 0.158, Intangible 0.192.

* Firm F E: Yes for all columns.

Moreover, comparing with the OLS estimates, we continue to observe the increased magnitudes of q in the GMM estimates, removing the downward bias of the OLS estimations. Specifically, in the GMM estimations, the magnitudes of the coefficient of q are 0.091, 0.060, and 0.034 for total, physical, and intangible investments, respectively. On the contrary, in the OLS regressions, the magnitudes of the coefficient of q are 0.028, 0.015, and 0.011, respectively. Therefore, our proposed GMM approach corrects the downward bias present in OLS estimation. From these results, we confirm that the results from our approach are robust across various types of investment.

F. Nonlinear Model Estimations

In this section, we examine the possibility that investment may respond nonlinearly to state variables. To explore this, we perform nonlinear regression analyses to evaluate the effectiveness of our proposed estimation approach for nonlinear models. We include higher-order terms such as lnK $ {}^2 $ , lnK $ {}^3 $ , and lnN_K $ {}^2 $ . The estimation results are reported in Table 7. In column 1, we incorporate both lnK $ {}^2 $ and lnN_K $ {}^2 $ into the regression. On the one hand, the coefficient of lnK $ {}^2 $ is significant at the 1% level, although the magnitude of 0.005 is relatively small. On the other hand, the coefficient of lnN_K $ {}^2 $ is not statistically significant. We find that $ \omega $ and q remain statistically significant at the 1% level. In column 2, we further check the nonlinearity of lnK by including its squared and cubic terms in the regression. While the coefficients of lnK $ {}^2 $ and lnK $ {}^3 $ have small magnitudes of −0.004 and 0.001, respectively, they are statistically significant at the 1% level. Regardless of the functional form of the investment equation, $ \omega $ and q are still statistically significant and economically meaningful. The coefficients of $ \omega $ and q are comparable to those in Table 3.

TABLE 7Nonlinear Estimations

TABLE 7 Long description

The table consists of three columns: Variables, 1, and 2.

* x: Column 1 is 0.023*** (16.87); Column 2 is 0.023*** (17.26).

* C F: Column 1 is 0.000 (0.19); Column 2 is 0.000 (0.02).

* ln K (size): Column 1 is -0.187*** (-12.79); Column 2 is -0.154*** (-12.34).

* ln K squared (size): Column 1 is 0.005*** (4.28); Column 2 is -0.004*** (-3.76).

* ln K cubed (size): Column 1 is empty; Column 2 is 0.001*** (8.47).

* ln N sub K (employment): Column 1 is 0.033*** (2.78); Column 2 is 0.022*** (5.05).

* ln N sub K squared (employment): Column 1 is 0.001 (0.99); Column 2 is empty.

* omega: Column 1 is 0.317*** (10.16); Column 2 is 0.323*** (10.36).

* Firm-year sample: 149,429 for both columns.

* e(Q): Column 1 is 0.00306; Column 2 is 0.00315.

* P-value of Hansen J: Column 1 is 0.44; Column 2 is 0.45.

* F stats of inv(t - 1): Column 1 is 220.06; Column 2 is 219.59.

* F stats of x(t - 1): Column 1 is 82.12; Column 2 is 82.07.

* F stats of x(t - 2): Column 1 is 165.93; Column 2 is 167.24.

* S W P of inv(t - 1): 0.00 for both.

* S W P of x(t - 1): 0.00 for both.

* S W P of x(t - 2): 0.00 for both.

* S W kappa squared P of inv(t - 1): 0.00 for both.

* S W kappa squared P of x(t - 1): 0.00 for both.

* S W kappa squared P of x(t - 2): 0.00 for both.

* Firm F E: Yes for both.

Therefore, even after accounting for the nonlinearity of investment in state variables, the dynamic coefficient of the persistent shock remains statistically significant. Overall, the empirical evidence supports the importance of controlling for the unobserved persistent shock and measurement error in q when estimating investment functions.